Bifurcation of periodic solutions in a nonlinear difference-differential equations of neutral type

Abstract

The existence of a self-sustained periodic solution in the autonomous equation \[ u ′ ( τ ) − α u ′ ( τ − h ) + β u ( τ ) + α γ u ( τ − h ) = ϵ f ( u ( τ ) ) u’\left ( \tau \right ) - \alpha u’\left ( {\tau - h} \right ) + \beta u\left ( \tau \right ) + \alpha \gamma u\left ( {\tau - h} \right ) = \epsilon f\left ( {u\left ( \tau \right )} \right ) \] is proved under appropriate assumptions on α , β , γ , f \alpha ,\beta ,\gamma ,f and h h . The method of proof consists in converting this equation into an equivalent nonlinear integral equation and demonstrating the convergence of an appropriate iteration scheme.